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In algebraic topology, we have the map lifting lemma which says that given a covering space $p:(\tilde{X},\tilde{x})\rightarrow (X,x)$ and a map $f:(Y,y)\rightarrow (X,x)$ with $Y$ connected and locally path-connected, a lift $\tilde{f}:(Y,y)\rightarrow(\tilde{X},\tilde{x})$ of $f$ exists if and only if $f_*(\pi_1(Y,y))\subset p_*(\pi_1(\tilde{X},\tilde{x}))$.

I want to ask whether analogues of the above are true for finite etale maps and etale fundamental groups.

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The answer is yes, at least for finite covers (or even pro-finite covers). The key is to carefully translate everything into the language of finite $\pi_1$-sets using the Galois correspondence. The canonical reference is of course SGA 1, Expose V - in particular $\S5,6$ and 7 once you're familiar with the basics.

By the universal property of fiber products, the existence of such a lifting $\tilde{f}$ is equivalent to the existence of a section of the covering map (ie finite etale morphism) $p_Y : Y\times_X\tilde{X}\rightarrow Y$ which contains the geometric point $(y,\tilde{x})$ (such a section if it exists, is unique).

By Galois theory, the pointed covering $p : (\tilde{X},\tilde{x})\rightarrow (X,x)$ is equivalent to the data of the pointed set $(p^{-1}(x),\tilde{x})$, together with the monodromy action of $\pi_1(X,x)$ on the underlying set $p^{-1}(x)$. The pullback $p_Y$ is (via the Galois correspondence) given by the pointed set $(p_Y^{-1}(y),(y,\tilde{x}))$ together with the monodromy action of $\pi_1(Y,y)$ acting on the underlying set, which is described by the map $$\pi_1(Y,y)\stackrel{f_*}{\rightarrow}\pi_1(X,x)\rightarrow Aut(p^{-1}(x))\cong Aut(p_Y^{-1}(y))\qquad (*)$$ where the last isomorphism is induced by the natural bijection $p_Y^{-1}(y)\cong p^{-1}(x)$.

(In short, the pullback of a cover over $X$ given by a $\pi_1(X)$-set $T$ by any morphism $Y\rightarrow X$ is the cover over $Y$ given by pulling back the $\pi$-set structure on $T$ via the morphism $\pi_1(Y)\rightarrow\pi_1(X)$. (What else could it be?) See $\S6$ in SGA 1 Expose V)

Under these identifications, $\pi_1(\tilde{X},\tilde{x})$ is naturally identified with the subgroup of $\pi_1(X,x)$ defined by $Stab_{\pi_1(X,x)}(\tilde{x})$. The map $p_Y$ admits a section containing the geometric point $(y,\tilde{x})$ if and only if the point $(y,\tilde{x})\in p_Y^{-1}(y)$ is fixed by the monodromy action of $\pi_1(Y,y)$ (Note that a section of $p_Y$ is a morphism of finite etale covers over $Y$, and hence corresponds via the Galois correspondence to a $\pi_1(Y,y)$-set morphism from the singleton set with trivial $\pi$-action to $p_Y^{-1}(y)$. Since this morphism is a morphism of $\pi_1(Y,y)$-sets, and hence must commute with the $\pi_1(Y,y)$-action, it implies that the image of this map is fixed by $\pi_1(Y,y)$).

However, by the definition of this monodromy action in $(*)$, we see that this is true if and only if $f_*\pi_1(Y,y)\subset Stab_{\pi_1(X,x)}(\tilde{x}) = \pi_1(\tilde{X},\tilde{x})$.

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    $\begingroup$ If I may: note that this gives a rather nice proof of the original statement in the topological case. $\endgroup$ – Piotr Achinger Jun 11 '18 at 19:29
  • $\begingroup$ @PiotrAchinger Hi Piotr! I guess I'm a little worried about the situation for infinite covers. Does the Galois correspondence (equiv. of categories between finite covers and finite pi-sets) extend to arbitrary covers? (iirc a Galois category as defined by grothendieck must have a fundamental functor taking values in finite sets) $\endgroup$ – Will Chen Jun 19 '18 at 13:12

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